Is there a use for discontinuity? In many theorems and applications, continuity is important to get certain results. So the idea is to get continuity in your solution.
Is there a reason you want discontinuity? 
 A: Not really - from a purely mathematical perspective, "discontinuity" is just the absence of a convenient property. If math were just about looking pretty, most mathematicians would just pretend everything was continuous.
However, discontinuous functions happen. It's like asking if there's some reason you'd want to have traffic jams - of course not, but they happen anyway, so we have to know how to deal with them. Discontinuities show up in applied math all the time; they even show up in "pure" (i.e., non-applied) situations, because there are some functions (e.g., $\sqrt{x}$) which are natural but inherently discontinuous.
A: In mathematics we impose conditions on objects which are some nice properties. We want to understand the behaviour and structure of those objects. Their  nicety conditions  make them  easier to analyse. 
But when mathematics is used to model real world along with nice things we also get some  not-so-nice things.
Let us steadily but slowly blow air into a balloon, its volume increases as the time goes by. The  volume of the balloon is a continuous function of time. But at one stage the balloon bursts,  causing break in continuity.
Even in commercial mathematics discontinuity occurs. Prize of 100 ml of some energy drink might be $\$x$.  So the cost of buying quntity $q$ is $qx$, a continuous function of $q$.  When you buy quantities higher than certain amount you get a discount, which again breaks continuity.
Taxation laws are also such that after certain income level there is jump in amount of tax. 
This is not only in calculus based mathematics. We study rational number. It is a nice condition. You can add or multiply rational numbers you again get a rational number.But then comes $\sqrt2$, as the length of the diagonal of a nice square of unit length, (a nice object of reality). But $\sqrt2$  is irrational; and irrationals when multiplied won't always produce irrationals.
There is another kind of wild numbers  not even half as decent as $\sqrt2$; they can't be obtained as solutions of any polynomial equation with integer coefficients  however  high the degree of  the polynomial we  choose. The number $\pi$ which is ubiquitous in physics and real world  is such a wild number.(Transcendental number).
Even though aesthetically pleasing mathematical edifice seem to work as good models we often encounter some tricky objects that forces us to treat discontinuities, singularities, divergent series/integrals, Dirac delta "functions" etc with more respect,
